(2019) Elasticity Tetrads, Mixed Axial-Gravitational Anomalies

PHYSICAL REVIEW RESEARCH 1, 023007 (2019)

Elasticity tetrads, mixed axial-gravitational anomalies, and (3+1)-d quantum Hall effect

J. Nissinen 1,* and G. E. Volovik1,2 1Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland 2Landau Institute for Theoretical Physics, Acad. Semyonov Av., 1a, 142432, Chernogolovka, Russia

(Received 4 June 2019; revised manuscript received 5 August 2019; published 6 September 2019)

For two-dimensional topological insulators, the integer and intrinsic (without external magnetic field) quantum Hall effect is described by the gauge anomalous (2+1)-dimensional [(2+1)-d] Chern-Simons (CS) response for the background gauge potential of the electromagnetic U(1) field. The Hall conductance is given by the quantized prefactor of the CS term, which is a momentum-space topological invariant. Here, we show that three-dimensional crystalline topological insulators with no other symmetries are described by a topological (3+1)-dimensional [(3+1)-d] mixed CS term. In addition to the electromagnetic U(1) gauge field, this term a a a contains elasticity tetrad fields Eμ(r, t ) = ∂μX (r, t ) which are gradients of crystalline U(1) phase fields X (r, t ) and describe the deformations of the crystal. For a crystal in three spatial dimensions a = 1, 2, 3 and the mixed axial-gravitational response contains three parameters protected by crystalline symmetries: the weak momentum- space topological invariants. The response of the Hall conductance to the deformations of the crystal is quantized in terms of these invariants. In the presence of dislocations, the anomalous (3+1)-d CS term describes the Callan- Harvey anomaly inflow mechanism. The response can be extended to all odd spatial dimensions. The elasticity tetrads, being the gradients of the lattice U(1) fields, have canonical dimension of inverse length. Similarly, if such tetrad fields enter general relativity, the metric becomes dimensionful, but the physical parameters, such as Newton’s constant, the cosmological constant, and masses of particles, become dimensionless.

DOI: 10.1103/PhysRevResearch.1.023007

I. INTRODUCTION tetrads, the elasticity tetrads have canonical dimensions of inverse length. As a result, the CS term is dimensionless (in The effective Chern-Simons (CS) description of the integer unitsh ¯ = 1), and as in the case of even space dimensions, (and fractional [1,2]) quantum Hall effect (IQHE) and the the prefactor is given by integer momentum-space topolog- ensuing topological quantization of Hall conductivity has ical invariants. The CS term leads to the analog of mixed been originally considered in even spatial dimensions [3]. (axial and gravitational/elastic) anomaly inflow in (3+1)-d Similarly, in (2+1)-dimensional [(2+1)-d] topological insu- (see, e.g., Refs. [19–23]). The structure of the CS term lators [4], in the generalized QED [5–7]orinthinfilmsof 3 reflects the Callan-Harvey mechanism of anomaly cancella- topological superfluids and superconductors [8,9], the intrin- tion [23], provided here by the fermion zero modes living sic or anomalous quantum Hall effect (AQHE) in the absence on dislocations/sample boundaries [24–26]. In this way, the of magnetic flux is also described by a CS term. For both CS response related to the (3+1)-d QHE is valid in the pres- IQHE and AQHE, the prefactor of this term is expressed in ence of deformations and satisfies the consistency conditions terms of momentum-space topological invariants: the Chern of gauge invariance and anomaly inflow. The same mecha- number(s). The same mechanism works for gapped systems nism works for gapped crystalline systems in all odd spatial in all even space dimensions [10–12]. dimensions. Here, we show that the IQHE/AQHE in (3+1)- The rest of this paper is organized in the following way. Be- dimensional [(3+1)-d] crystalline quantum Hall systems or fore introducing the elasticity tetrads and the ensuing effective topological insulators [13] is also described by CS term response in (3+1)-d in Secs. III and IV, we first review the with mixed field content. Namely, the (3+1)-d CS term simpler (2+1)-d QH case. Section V describes the anomaly features elasticity tetrads [14–17], which describe the geom- inflow mechanism of the action in (3+1)-d. In Sec. VI,we etry of elasticity theory, including crystals with dislocation briefly describe the extension to arbitrary even space-time defects. The density of dislocations corresponds to spatial dimensions. The coupling of the elasticity tetrads to space- torsion of the geometry, more familiar in gravitational the- time geometry in QHE is discussed in Sec. VII along with the ories (see, e.g., Ref. [18]). In contrast to the gravitational speculative possibility to have quantized and dimensionless gravitational couplings if the gravitational space-time metric is identified with the metric of elasticity tetrads. We conclude *Corresponding author: jaakko.nissinen@aalto.fi in Sec. VIII.

Published by the American Physical Society under the terms of the II. (2+1)-d TOPOLOGICAL ACTION FOR QHE Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) The topological Chern-Simons action for the IQHE and for and the published article’s title, journal citation, and DOI. the anomalous, intrinsic (i.e., without external magnetic ﬁeld)

2643-1564/2019/1(2)/023007(9) 023007-1 Published by the American Physical Society J. NISSINEN AND G. E. VOLOVIK PHYSICAL REVIEW RESEARCH 1, 023007 (2019)

AQHE in the D = (2+1)-dimensional crystalline insulator is δλAμ = ∂μλ: given by [4,6–8] 1 3 μνλ δλS A =− d x λ ∂μσ∂ν Aλ CS[ ] π 1 2 ναβ 4 S + Aμ = N d xdt Aν ∂αAβ , (2 1)-d[ ] π (1) 4 = 3 λ∂ μ =−δ . d x μ jH λSbndry[A] (7) where |e|=h¯ = 1 and the electromagnetic U(1) gauge field Aμ has dimensions of momentum. The integer prefactor N The anomalous divergence is compensated by protected edge in the response is expressed in terms of momentum-space modes on the boundary described by Sbndry. topological invariant [3,6–8] III. ELASTICITY TETRAD FIELDS 1 ∞ N = ij dω dS The pure (3+1)-d CS term cannot be defined in even 8π 2 −∞ BZ space-time dimensions, therefore, the QH response requires × ∂ −1 ∂ −1 ∂ −1 , Tr (G ωG ) G ki G G k j G (2) additional fields and constitutes a mixed response which is only weakly topologically protected. These fields are due to where the spatial momentum integral is over the 2-d torus of the (weak) crystalline symmetries of the system. the two-dimensional Brillouin zone (BZ). The integer N is a Specifically, these are the elastic deformations of the a topological invariant of the system and in particular remains crystal lattice described in terms of elasticity tetrads Eμ(x), locally well defined under smooth deformations of the lattice. which represent the hydrodynamic variables of elasticity Under sufficiently strong deformations or disorder one can theory [14,15]. In the absence of dislocations, the tetrads a a μ have regions of different N(x) with the associated chiral edge E = Eμdx are exact differentials. They can be expressed modes. In that case, the global invariant, if any, is defined in general form in terms of a system of three deformed by the topological charge of the dominating cluster which crystallographic coordinate planes, surfaces of constant phase percolates through the system [27]. X a(x) = 2πna, na ∈ Z, with a = 1, 2, 3, in three dimensions. The intersections of the three constant surfaces A. (2+1)-d bulk Chern-Simons and consistent X 1(r, t ) = 2πn1, X 2(r, t ) = 2πn2, X 3(r, t ) = 2πn3 (8) boundary anomaly are points of the (possibly deformed) crystal lattice The topologically protected physics of the quantum Hall ={ = , , | ∈ 3, a ∈ 3}. effect arises due to the Callan-Harvey [23] anomaly inflow L r R(n1 n2 n3) r R n Z (9) of the Chern-Simons action from the bulk to the bound- The elasticity tetrads are gradients of the three U(1) phase ary [28–30], which we now review. In fact, the boundary fields X a, a = 1, 2, 3, μ + current Jbdry realizes the (consistent) (1 1)-d chiral anomaly a a Eμ(x) = ∂μX (x) (10)

μ N μν bdry and have units of crystal momentum. By the inverse function ∂μJ = Fμν , (3) bdry 8π theorem, we can define the inverse vectors ν μ 0, bdry a = δ . , = ∂ − ∂ Eμ(x)E a(x) ν (11) where Jbdry F0 t A At are along the boundary. In this way the protected edge modes arise from the cancellation In the simplest undeformed case, X a(r, t ) = Ka · r, where of bulk and boundary gauge anomalies [7,23,28,29]. In more (0)a ≡ a a Ei K are the (primitive) reciprocal lattice vectors K . detail, the (2+1)-d Chern-Simons term (1) can be written as In the general case, they depend on space and time and are quantized in terms of the lattice L in Eq. (9). 1 3 μνλ a SCS[A] = d x σ (x)Aμ∂ν Aλ, (4) In the absence of dislocations, when X (x) are globally π a 4 well defined, the tetrads Eμ(x) are pure gauge and satisfy the integrability condition (in differential form notation) where σ (x) = N (x1) has a step function domain wall at a a 1 a a μ ν 1 = = = ∂μ − ∂ν ∧ = . x 0. This is equivalent to a space-time manifold with a T dE 2 Eν (x) Eμ(x) dx dx 0 (12) boundary. The current is In the presence of dislocations, T a = 0, and X a(x) are multi- δ σ valued. μ = SCS[A] = μνλ∂ − 1 μνλ ∂ σ jH ν Aλ ( ν )Aλ (5) We can also consider a metric associated with these tetrads δAμ 2π 4π a b gμν = EμEν ηab, (13) with bulk and boundary contributions. We see that the Hall 2 where η is the metric associated to the background lattice, conductivity is σH = Ne /h. The current has the divergence say the spatial Euclidean or Minkowski metric. The important 2 μ ν = μν μ 1 μνλ N 1 1νλ N difference is that dn g dx dx in terms of the elasticity ∂μ j = ∂μσ∂ν Aλ = δ(x ) Fνλ = E , (6) H 4π 8π 4π tetrads is dimensionless and therefore counts the space-time distances in terms of the lattice points of L [16]. The tetrad a where we have specialized to the case when F02 =−E is fields Eμ, but not the metric, enter the CS action for a (3+1)-d an electric field on the boundary. This arises because of quantum Hall effect of the lattice QH systems and topological the consistent gauge anomaly under gauge transformations insulators as we now discuss.

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IV. (3+1)-d TOPOLOGICAL ACTION FOR QHE Now, we want to obtain the effective action to first order in gradients of Aμ(x). We use the fact that the coordinate Using the elasticity tetrads, the generalization of the dependence of Aμ(x) is semiclassical, i.e., the field varies (2+1)-d QH response to (3+1)-d is in principle straightfor- + slowly on the scale of the lattice momenta. In the semi- ward. This extends the electromagnetic (2 1)-d CS response ≡ ≡ + classical phase-space analysis [6,8,11,38], with G G[A] to (3 1)-d with the following topological terms, featuring ,ω , a G(p ; Aμ(x t )), we obtain to the lowest order in the gradi- U(1) fields in combination with the elasticity tetrads Eμ(x): ents of Aμ(x) 3 1 4 a μναβ 3 ω S + [Aμ] = Na d xEμ Aν ∂αAβ , (14) i d p d − − (3 1)-d 2 =− 3 ∂ μ 1 ∂ 1 8π Seff [A] d x dttr G x G G kμ G a=1 4 (2π )4 −1 −1 −1 a = , , − ∂ ∂ μ ∂ , where 1 2 3 labels the spatial lattice directions in three G kμ G G x G G kν G | Aν (17) space dimensions. The derivation of this formula using semi- A=0 a μ classical expansion is in the next section. The tetrads Eμ(x)in where x = (t, x) and kμ = (ω, −p) and Eq. (10) have the dimension of the momentum, and thus the = − − integrals in Eq. (14) are dimensionless (¯h 1).Itfollows,as ∂ μ 1 = ∂ 1 ∂ μ . G x G G kν G| x Aν (18) in the (2+1)-d case, that the prefactors are dimensionless and A=0 are also expressed in terms of integer topological momentum- The trace in the matrix product of Green functions is antisym- space invariants. The integer coefficients N are antisymmetric a metric in the indices of kμ derivatives. With the convention integrals of the Green’s functions: txyz =−1, we arrive to ∞ 1 i Na = ijk dω dS 3 ω π 2 a i 3 αβγρ d p d 8 −∞ BZ S [A] = d xdt Aα∂β Aγ μνλρ eff π 4 −1 −1 −1 12 (2 ) × Tr (G∂ωG ) G∂ G G∂ G , (15) ki k j × ∂ −1 ∂ −1 ∂ −1 . tr G kμ G G kν G G kλ G A=0 (19) where the momentum integral is now over the restricted 2-d i a BZ torus determined by the area dSa normal to Ei . The momentum-space prefactor can be separated to be of the A similar expression for the (3+1)-d QH was proposed in form Ref. [31] without the elasticity tetrads and space-time depen- + d2p dω dence. In the deformed crystalline systems in (3 1)-d, the aδρ a i dp a μνλρ tetrads Eμ(x) entering the (3+1)-d Chern-Simons action (14) 24π 2 depend slowly on space and time. For arbitrary background × ∂ −1 ∂ −1 ∂ −1 tr G kμ G G kν G G kλ G fields, the space-time dependence violates the gauge invari- A=0 ance of the action. However, in the absence of dislocations the = a a = a , latter does not happen for the elasticity tetrads: under defor- dp Na(p ) K Na (20) mations Eq. (14) remains gauge invariant due to the condition a = δ δ = ∂ φ dE 0inEq.(12). The variation φS under Aμ μ where Na = Na(pa )isthe(2+1)-d integer-valued invariant, / is identically zero modulo the bulk boundary QH currents 2 ω of the sample. The anomaly cancellation in the presence of a d p d N (p ) = μνλ dislocations is discussed in detail below. a a 24π 2 Here is the main difference between the topological insu- × ∂ −1 ∂ −1 ∂ −1 lator and a gapless system in (3+1)-d, e.g., a Weyl semimetal. tr G kμ G G kν G G kλ G A=0 (21) Integrating out the fermions leads to the chiral anomaly in the bulk and the ensuing gauge anomaly has to be consistently evaluated in imaginary time over the cross-sectional recipro- a canceled. In that case, instead of the elasticity tetrads the cal space, perpendicular to the reciprocal lattice direction K . four-momentum separation of the Weyl points appears in The elementary cell can be taken to be triclinic. Topologically, a the counterterm [21,32–37]. In the deformed system, this the integration is over a (pinched) 3-torus and Na(p )isan π , = parameter depends on the coordinates, but the integrability element of 3(GL(n C)) Z [6]. condition is now absent, and the term cannot arise without We conclude that the bulk chiral gauge anomaly. 1 3 μνλρ (0)a S [A] = N d x dt Eμ Aν ∂λAρ , (22) eff 8π 2 a A. Semiclassical expansion (0)a a i a The response in Eq. (8) can be obtained via the semi- where Eμ = (K )iδμ = dp , i = x, y, z, are the reciprocal classical Green function formalism [6,8,38]. The assumptions lattice vectors normal to the different lattice planes with are gauge invariance and semiclassical expansion to lowest invariants Na. This is the statement that the (3+1)-d QHE is (0)a order. The effective action for a slowly varying background described by the weak vector invariant NaEμ protected by field Aμ(x, t )is the crystalline symmetry. It is well known that the topological winding number Na Z[A] G[A] is stable against small variations δG(k , k , k ) of the Green’s log = iSeff [A] = i tr ln . (16) x y z Z0 G0 function that do not close the gap [6,8,11]. We can consider

023007-3 J. NISSINEN AND G. E. VOLOVIK PHYSICAL REVIEW RESEARCH 1, 023007 (2019) small lattice deformations xμ → xμ + ξ μ(x), ξ 1. Under Eq. (12), and the current represents a dissipationless, fully these, the reciprocal vectors change as reversible current, which is conserved due to U(1) gauge invariance E (0)a = dpa μ μ μ ∂μJ = ∂μJ = 0. (29) bulk π a a 2 a a → Eμ(x) ≡ dp (x) ≈ δμ − ∂μξ , (23) d C. Chiral magnetic effect a where Eμ(x) is a semiclassical, slowly varying field at the The action (14) and current (28) also describe the chiral lattice scale d in reciprocal space which defines the local magnetic effect (CME) [41,42]. In the presence of periodic normal direction of a set of lattice planes. Note that the directions varying in time, time dependence X a(r, t ) appears. topological invariants Na are assumed to be constant and In the CME an electric current along an applied magnetic field independent of deformations throughout. The final result is is induced: Eq. (8). 1 3 J = N E aB. (30) 4π 2 a t B. Hall current in terms of elasticity tetrads a=1 a = ∂ a/∂ The elasticity tetrads, i.e., elementary deformed reciprocal This current contains Et X t and thus it vanishes in a a and direct lattice vectors Eμ(x) = ∂μX (x) and the inverse equilibrium in agreement with Bloch theorem, according to μ which the total current is absent in the ground, or in general Ea (x) appear in the EM response, descending from the lattice a field phase fields NaX (x) in the presence of deformarions. As any equilibrium, state of the system (see, e.g., Ref. [43]). discussed, the phase fields satisfy the quantization condition Here, we restrict to spatial lattices under deformations. The 0 = ω δ0 X a(x) = 2πna, na ∈ Z3, on the lattice points x ∈ L.TheHall CME for a time-periodic insulator with timelike Eμ F μ ω = conductance is [39] (with Floquet drive F ) and the temporal invariant N0 0 was pointed out in Ref. [17]. This can be extended to spatial N E a(x) σ = a a k , (24) deformations as well. ij ijk 4π 2 i.e., the conductance is quantized in planes perpendicular to V. ANOMALY CANCELLATION AND DISLOCATION = a the layer normal Gi a NaEi (x). The reciprocal vector ZERO MODES G = Gi is the weak Hall index of a weak (3+1)-d Chern insulator. A. Callan-Harvey effect on dislocations and mixed anomaly (0)a In the nondeformed crystal, where Ek are primitive Now, we describe the anomaly inflow. The constraint (12) reciprocal lattice vectors, this equation transforms to the is violated in the presence of topological defects: dislocations. well-known equation with (3+1)-d quantized Hall conductiv- The density of dislocations equals the torsion for the elasticity ity [13,32,36,39]: tetrads (with vanishing spin connection): 2 a = a, a = ∂ a − ∂ a , , = , , e T dE Tkl kEl l Ek k l x y z (31) σij = ijkGk, (25) 2πh similar to the role of space-time torsion in gravitational where Gk is a reciprocal lattice vector, which is expressed theories [14,18]. The nonzero dislocation density or torsion in terms of the topological invariants Na and the primitive violates the conservation of the Hall current: (0)a = a reciprocal lattice vectors E K : − 3 μ 1 1 μναβ a ∂μJ = Fαβ N Tμν . (32) 3 8π 2 4 a = (0)a. a=1 Gk NaEk (26) a=1 This mixed anomaly represents the Callan-Harvey mechanism In the deformed crystal, the conductivity tensor is space- of anomaly cancellation [23], which is provided here by time dependent, and thus is not universally quantized. the fermion zero modes living on dislocations [23–26,40]. However, the response of the conductivity to deformation is The action remains well defined and gauge invariant in the quantized: presence of dislocations, i.e., 2π ambiguities in the phase field X a, due to zero modes with (1+1)-d covariant anomaly along σ 2 d ij e the dislocation string, = ijkNa. (27) a π μ μ μ dEk 2 h ∂ = ∂ + ∂ = , μJ μJbulk μJdislocation 0 (33) The Hall current is as shown in Refs. [23,40,44] for Dirac fermions in the pres- − 3 μ 1 μναβ a ence of complex vortexlike axionic mass. The Dirac model J = Na Eν ∂αAβ + 4π 2 can be taken as a topological model for the (3 1)-d QH a=1 system [11], which we discuss next. 3 1 μναβ a + N ∂αEβ Aν (28) B. Dirac fermion model 8π 2 a a=1 To verify the existence of dislocation zero modes on and has a bulk and a topological defect/boundary compo- dislocations and overall gauge invariance, let us consider a nent [23,40]. In the absence of dislocations, dEa = 0in simplified model with the same symmetries as the

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iγ 5X (x) ∂ X The complex scalar m(x)e is equivalent to a slowly λ varying mass profile. A vortex line singularity (“axionic string” [23,40]) is the profile where m(x) → m far from the p + q λ p + q + k singularity and m(xc ) = 0, where xc is the vortex location. The phase field X (x) has a vortex singularity q μ ν q dX = 2πn (35) C

around all contours around xc and is is such that X = const + a p nφ far from the line defect. In the present case, X = NaX a and n = n Na. The expansion in Eq. (34) applies far from the FIG. 1. The four-dimensional gauge ﬁeld vacuum polarization vortex. Let us compute the induced effective action far from μν (q, k)inEq.(38) with the axionic vertex ∂λX. the vortex string by starting the from the expansion iγ 5X μ 5 time-reversal and parity-breaking topological insulator with mψ e − 1 ψ(y) = im∂μX (y − x) ψγ ψ(x) +··· . crystalline order, namely, a gapped and time-reversal and parity-breaking Dirac model with a complex mass induced by (36) a scalar ﬁeld X, The response is found from the axial bosonic polarization μν S[ψ,ψ,Aμ, X] vertex (x, y)[40] (or the equivalent Goldstone-Wilczek current [44]) of the effective action 3 μ iγ 5X μ = d x dt ψ(γ i∂μ)ψ − m(x)ψe ψ − ψγ ψAμ 4 4 1 μν Seff [A] = d xd yAμ(x) (x, y)Aν (y) (37) 3 μ 2 = d x dt ψ(γ i∂μ − m)ψ μ ν λ ρ depicted in Fig. 1 becomes, using tr[γ 5γ γ γ γ ] = iγ 5X μ μνλρ txyz −mψ e − 1 ψ − ψγ ψAμ +··· . (34) +4i with =−1, 4 4 ∂ μν d qd p 4 (4) iq(y−x) μ ν 5 (x, y) = m∂λX d k δ (k)e tr[iγ G(p)iγ G(p + q)γ G(p + q + k) + (μ ↔ ν, k ↔ q)] (2π )8 ∂kλ 4 4 2 μνλρ d qd p iq(y−x) m iqρ 1 1 =+4i ∂λX e + . (38) (2π )8 (p2 − m2 )[(p + q)2 − m2] p2 − m2 (p + q)2 − m2

Taking the limit q → 0, we arrive to the bulk effective ac- The induced current separates to bulk and boundary contri- tion [23,40] butions 1 4 μνλρ a μ S [A, X] = N d x ∂μX Aν ∂λAρ +··· J =δSeff /δAμ eff a 8π 2 a −1 μνλρ 1 μνλρ = ∂ν X∂λAρ + (∂λ∂ν X )Aρ (40) (39) 4π 2 8π 2 z after integrating out the massive fermions far from the string. due to an electric field F0z =−E parallel along the string. The response follows from Eq. (38) by the replacement X → This has contributions from the bulk, where X (r,φ,z) = nφ, a a NaX , i.e., instead of a single U(1) axion field, we have a and from the string where, from Eq. (35), we get flavored U(1)3 lattice phase field X a with topological charges −tijz∂ ∂ = π δ(2) , , = , . Na which are not protected [38] without the existence of the i jX 2 n (xc ) i j x y (41) a slowly varying lattice phase field X . Therefore, Let us now extend the above effective action (39)tobe − valid everywhere, essentially by considering a vortex string μ 1 μνλρ J = ∂ν X∂λAρ with a delta-function core. bulk 4π 2 1 n 1 n = F =− E z = Jr , (42) C. Induced anomaly current 4π 2 r 0z 4π 2 r bulk The bulk effective action (39) was derived far from the whence the current per unit time and length in the radial string defect. Let us now check the induced current due to a direction outward from the string is − n E z. In addition, there = a 2π topological configuration in X (x) NaX (x). For simplicity, is the current localized on the string we will consider singularities only in the slowly varying con- a tinuum lattice field X (x) and treat the topological numbers i 1 μνλρ J = (∂λ∂ν X )Aρ string 2 Na as constants. A dislocation in the topological response is 8π πnaN X = N X a x na a2 a ambiguity in the field a ( ), and is the 1 (2) ij n ij (2) = a = (2πnδ (xc )) A j = A jδ (xc ), (43) Burgers vector of the dislocation and n n Na. 8π 2 4π

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, = , ∂ i = n ij δ(2) = Variation of Eq. (47) gives the electric current i j t z, with the divergence i Jstring 8π Fij (xc ) n zδ(2) N π E (xc ). The divergence of the total current (40)onthe μ 1 μν 1 4 J = Eν , (49) string is anomalous 2π the conservation of which μ 1 μνλρ μ a 1 ∂μ J =− (∂μ∂ν X )Fλρ ∂μJ = N dE = N dE = 0 (50) 8π 2 a 1 has a simple interpretation. The condition dEa = 0 is equiv- 1 n = (2πnδ(2)(x ))F =− E zδ(2)(x ). (44) alent to the conservation of the sites of the one-dimensional 4π 2 c 0z 2π c lattice, whereas the index N1 corresponds to the number of the On the other hand, the chiral fermions on the string, where electrons per site, which is integer for band insulators. As a they are effectively massless, result in n chiral zero modes result, the number of the electrons is trivially conserved under with chiral anomaly adiabatic deformations. Since any one-dimensional insulator is described by the ∂ i = ∂ i + ∂ i,cons. i J(1+1)-d i Jstring i J(1+1)-d topological invariant N1, which can only change when the z gap closes, we may call any (1+1)-d insulator topological, 1 ij F0z E =± Fij =∓ =± . (45) although the topology of the filled states can only be detected 4π 2π 2π by higher invariants: the Chern numbers. In fact, this is This is the (1+1)-d covariant anomaly and is composed of the very similar to ordinary metals with Fermi surfaces [11]: boundary consistent anomaly plus the contribution localized the gapless Fermi surface represents a topological object in on the string from the bulk [21,40]. The anomaly due to the momentum space protected by an invariant similar to N1. bulk-induced current and zero modes cancels. Topology provides the stability of the Fermi surface with re- In contrast, from the previous arguments in Sec. II A we spect to interactions and explains why metals can be described seethatthe(2+1)-d QH anomaly is matched by the consistent by Landau Fermi-liquid theory. In this sense, metals can be anomaly of N-boundary (1+1)-d chiral fermions. This seems considered as topological materials, making the zeroth-order to coincide with the observation in Ref. [21]: The covariant invariant N1 one of the most important topological invariants anomaly is a Fermi-surface effect, whereas the consistent in the hierarchy of the topological invariants for fermionic anomaly arises from the contribution of all the filled levels. systems. In particular, it gives rise to the Luttinger theorem: the number of states in the region between two Fermi points + VI. EXTENSION TO EVEN D = 2 + 2n in (1 1)-d does not depend on interaction and arises through SPACE-TIME DIMENSIONS topology and adiabatic evolution. This can be generalized for any closed Fermi surface or insulator in higher dimen- The anomaly equation (32) can be straightforwardly ex- sions [11,45,46]. tended to arbitrary even D = 2 + 2n space-time dimensions: VII. GRAVITATIONAL QH RESPONSE AND EMERGENT GRAVITY 2n+1 μ a ∂μJ ∝ NaT ∧ F ∧···∧ F . (46) Now, we briefly describe the anomalous coupling of the a=1 n times elasticity tetrads to the effective space-time metric in the quantum Hall system and speculate on the possibility of a = a In addition to the torsional field strength T dE , it con- relating the elastic metric with the gravitational space-time tains the n-fold antisymmetric tensor product of U(1) gauge metric in an effective low-energy effective model of quantum field strengths F = dA, while the integer-valued topological + gravity. invariants Na are expressed in terms of (2n 1)-dimensional The dimensional elasticity tetrads in the (3+1)-d QHE integrals in frequency-momentum space [7]. + = allow us to write the dimensional extension of the (2 1)-d Equation (46) is valid even in case of n 0, i.e., in one gravitational framing anomaly as a mixed elastic-gravitational spatial dimension, producing the expected results. Consider anomaly in (3+1)-d [23,47–50]: a gapped one-dimensional chain of electrons with the action 3 N (see also [37]) = a 3 a ∧ μ ∧ ν Seff,g d xdtE ν d μ 192π 2 N1 μν 1 a=1 S + A = dxdt E Aν , (1 1)-d[ ] π μ (47) 2 2 μ ν ρ + ν ∧ ρ ∧ μ , (51) where the index N1 is defined via the Green’s function 3 ω, μ μ λ G( kx ), where N are effective central charges, ≡ dx are a ν λν the Christoffel symbol one-forms of the space-time metric, 1 −1 N1 = dω TrG(kx,ω)∂ωG (kx,ω). (48) that arises for example through Luttinger’s argument [51]. 2πi In particular, the mixed elastic-gravitational Chern-Simons The index N1 is the same for any kx without gap closings. term (51) implies the generalization of the (2+1)-d thermal Note that in (3+1)-d the index Na in Eq. (15)isthesamefor Hall effect [49,51–54]to(3+1)-d quantum Hall systems any cross section Sa of the three-dimensional Brillouin zone, and topological insulators with intrinsic Hall effect, although while in (1+1)-d the cross section corresponds to one point kx it is third order in derivatives and therefore beyond linear in the one-dimensional Brillouin zone. response.

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Finally, let us speculate on the hypothetical connection have shown how gauge invariance and anomaly inflow arise in a of the elasticity tetrads Eμ to real space-time metric and the presence of deformations of the weak lattice symmetries gravity. While the microscopic structure of the relativistic protecting the state. The response is a mixed CS response quantum space-time vacuum is not known, phenomenological featuring the elasticity tetrad fields of the lattice. This mixed approaches and effective field theory can be used to describe and geometric nature of the (3+1)-d CS response is to be the effects of the vacuum degrees of freedom. In one of these contrasted with the more familiar topological BF theory in scenarios, it is assumed that the space-time vacuum has the four dimensions (see, e.g., Ref. [59]), which represents a properties of a (3+1)-d superplastic crystalline medium con- mixed response with two one-form gauge fields and their the a structed from Eμ [55]. As in the condensed matter elasticity field strengths appearing in the action. theory, dislocations and disclinations in this space-time crystal More generally, the dimensionful elasticity tetrads are describe torsion and curvature of general relativity [14,56– the proper hydrodynamic variables related to the weakly 58]. More specifically, in this superplastic model of gravity, protected IQHE/AQHE/CME on general even-dimensional the size of the elementary cell in the vacuum space-time crystalline backgrounds. They describe the topological QH re- crystal is not fixed but in principle can vary arbitrarily with sponse with elastic and geometric deformations in systems in no elementary Planck scale space-time lattice. As a result, the odd spatial dimensions. In general, Eq. (32) and its extension induced action for the gravitational field [55] to higher dimensions, Eq. (46), describe the mixed anomaly in terms of the gauge fields and elastic torsion. These equations a 4 S Eμ = d x |E| (KR + ) (52) do not contain any parameters, except for topological quantum M3,1 numbers. This is because the elastic tetrads (torsion) have the contains only dimensionless quantities. Here, the metric canonical dimensions of [l]−1 (respectively [l]−2) instead of a b 0 −1 gμν = ηabEμEν is from Eq. (13) and the Ricci curvature scalar the conventional l (respectively [l] ) for the gravitational a R[Eμ] depends on the elasticity tetrads in the standard way, space-time tetrads (torsion). This allows one to write many thus making the gravitational constant K (inverse Newton mixed “quasitopological Chern-Simons terms,” analogous to constant) and the cosmological constant dimensionless, mixed axial-gravitational/elastic anomalies, in (3+1)-d quan- [K] = [R] = [] = 1. The same holds for higher derivative tum Hall systems with weak crystalline symmetries. In princi- gravity and for the other (nongravitational) physical quanti- ple, we can envisage similar geometric extension of BF theory ties, such as particle mass [M] = 1. Therefore, if gravity is to (4 + 1)-d using elasticity tetrads. related to fundamental elasticity tetrads, all the measurable physical quantities are dimensionless with respect to the space-time lattice. ACKNOWLEDGMENT This work has been supported by the European Research VIII. CONCLUSION Council (ERC) under the European Union’s Horizon 2020 In this paper, we have described the mixed topological research and innovation programme (Grant Agreement No. response of (3+1)-d QH systems using elasticity tetrads. We 694248).

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